Seminars and Colloquia by Series

Dynamical critical 2d first-passage percolation

Series
Stochastics Seminar
Time
Thursday, March 3, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
David HarperGeorgia Tech

In first-passage percolation (FPP), we let \tau_v be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of \tau_v, there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results which show that if sum of F^{-1}(1/2+1/2^k) diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of k^{7/8} F^{-1}(1/2+1/2^k) converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP. This is a joint work with M. Damron, J. Hanson, W.-K. Lam.

This talk will be given on Bluejeans at the link https://bluejeans.com/283104959/2281

The Distribution of Class Groups for Number Fields

Series
Job Candidate Talk
Time
Monday, February 21, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
online
Speaker
Jiuya WangUniversity of Georgia

Class group is a central object of the study in number theory. We will give an overview of our understanding of the distribution of class groups, with an emphasis on recent progress. In particular, we will explain how the use of symmetry in multiple ways turns out to be an essential tool to obtain many new results on the distribution of class groups.

https://bluejeans.com/555956601/7083

Zarankiewicz problem, VC-dimension, and incidence geometry

Series
Job Candidate Talk
Time
Thursday, February 17, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/939739653/6882
Speaker
Cosmin PohoataYale University
The Zarankiewicz problem is a central problem in extremal graph theory, which lies at the intersection of several areas of mathematics. It asks for the maximum number of edges in a bipartite graph on $2n$ vertices, where each side of the bipartition contains $n$ vertices, and which does not contain the complete bipartite graph $K_{s,t}$ as a subgraph. One of the many reasons this problem is rather special among Turán-type problems is that the extremal graphs in question, whenever available, always seem to have to be of algebraic nature, in particular witnesses to basic intersection theory phenomena. The most tantalizing case is by far the diagonal problem, for which the answer is unknown for most values of $s=t$, and where it is a complete mystery what the extremal graphs could look like. In this talk, we will discuss a new phenomenon related to an important variant of this problem, which is the analogous question in bipartite graphs with bounded VC-dimension. We will present several new consequences in incidence geometry, which improve upon classical results. Based on joint work with Oliver Janzer.
 

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